Simple, accurate, and efficient revisions to MacCormack and Saulyev schemes: High Peclet numbers

Stream water quality modeling often involves numerical methods to solve the dynamic one-dimensional advection–dispersion–reaction equations (ADRE). There are numerous explicit and implicit finite difference schemes for solving these problems, and two commonly used schemes are the MacCormack and Saulyev schemes. This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Peclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank–Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes were tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances were considered with respect to stability, accuracy, and computational efficiency. Results indicated that both the modified Saulyev and the MacCormack schemes, which are named the Saulyevc and MacCormackc schemes respectively, greatly improved the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyevc scheme. The MacCormackc scheme demonstrated increased time consumption but was still much faster than implicit schemes.